A Nice Polynomial Equation | Cubic Formula?

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PLAYLISTS 🎵 :

[larry]

As someone commented before, rational root theorem and synthetic division helps you find and verify -1 as a factor, but not only that, synthetic division, gives you the next factor, x^2+x-1. Solving this quadratic gives you the same complex solutions as in the video.

[renedelatorre]

In the second method, use synthetic division. It is a lot faster.

[trumpetbob]

Have an error at the 1:52 mark. Need to multiply 4/9 by 2 first before subtracting 1. It wouldn't be -10/9; it would be -1/9 but since you didn't finish that method, it didn't affect the solution.

[NadiehFan]

There is an error at 1:52. You should have gotten

t³ − ⁴⁄₃t = ¹¹⁄₂₇

This cubic equation has three distinct real roots since (¹¹⁄₅₄)² + (−⁴⁄₉)³ = −⁵⁄₁₀₈ < 0. Therefore, using the cubic formula(s) will only let you express the solutions with cube roots of complex numbers even though all solutions are real.

Of course you can still use the rational root theorem here to search for a rational solution, but that defies the whole exercise of depressing the cubic because you can just as well or even better apply the rational root theorem to the original equation

x³ + 2x² = 1

[lawrencejelsma]

I simply used Synthetic Division. With -1 on the left of 1 2 0 -1 the 1 slides down in column 1 for x^2, (1)(-1) + 2 = 1 slides down for x. (1)(-1) + 0 = -1 slides down for -1. Lastly, (-1)(-1) + -1 = 0 shows no remainder in the last column. We then have (x + 1)(x^2 + x - 1) easily by Synthetic Division.

Synthetic Division is so easy of a thought process than overcomplicating the problem by a^3 + b^3 = a^3 + 3ab(a + b) + b^3 and tricky t substitution extra thought processes not required in this problem.

[barberickarc]

Recently I've gotten in to solving 3rd order differential equations with variation of parameters. They take a bit more time and effort but are very satisfying, could you solve one for fun? :)

Here's some example problems I've done you could look at

y''' + 6y'' + 12y' + 8y =12*e^(-2x)

y''' - 9y'' + 15y' + 25y =x


The first one has nicer solutions; it would really make my day to see you cover this beautiful branch of higher mathematics, there's sadly not much information or content covering it here on YouTube

[rakenzarnsworld]

x = -1
(x+1)(x^2+x-1)=0
x = {-1+-5^(1/2)}/2

[pedrovargas]

-1, just by factorizing the left-hand side. But, actual resolution... 🤯

[vladimirkaplun]

May Niccolò Tartaglia and Gerolamo Cardano expect some bonus from you for using their method in every third video :) ?

[bjorntorlarsson]

Is it as a student generally allowed to "solve by inspection", or does one risk being whipped by the teacher like Carl Gauss? Academic maths papers can be hard to read because of phrases like "So, obviously:" But as long as it adds up it's just fine at that level.

[bjorntorlarsson]

I think I get it. After finding one root, one wants to factor it out because then the remaining factor must also be a root (i.e. equal 0).

[scottleung]

Got 'em all using the second method.